# Permutation Polynomials and Orthomorphism Polynomials of Degree Six

A classic paper of Dickson claims to give a complete list of
permutation polynomials of degree less than 6 over arbitrary finite
fields, and degree 6 over finite fields of odd
characteristic. However, recent published statements have suggested
that Dickson's classification was incomplete in the degree 6
case. We uncover the reason for this confusion, and establish the true
list of degree 6 permutation polynomials over all finite fields. As
an application, we determine the complete list of degree 6
orthomorphism polynomials. Additionally, we note that one of the
derived families of permutation polynomials provides a counterexample
to a published conjecture of Mullen.